Mixing by piecewise isometry in granular media

Transcription

1 in granular media Department of Mathematics University of Leeds PANDA, 16 September 2008 Leeds Joint work with Steve Meier, Julio Ottino, Northwestern Steve Wiggins, University of Bristol

2 Mixing Mixing of granular materials: is important Science 125th anniversary identified granular flow as one of the 125 big questions in Science is ubiquitous pharmaceuticals, food industry, ceramics, metallurgy, construction was initially explained by analogies with fluid mixing hence terms like granular shear and granular diffusion But the big difference is that granular materials tend to segregate

3 Segregation Granular materials segregate by (at least) 2 mechanisms: Percolation little particles fall through the gaps of big particles Buoyancy less dense particles tend to rise The Brazil Nut effect

4 Tumbler Mixers

5 Different tumbler geometries

6 Flow regimes [from S. W. Meier et al., 2007]

7 Tumblers [from S. W. Meier et al., 2007]

8 [Zuriguel, I., Gray, J.M.N.T., Peixinho, J. & T. Mullin (2006). Phys. Rev. E 73, ]

9 2D circular tumblers In the bulk ṙ = 0, θ = ω In the flowing layer ẋ = γ(δ(x)+y), ẏ = ωxy/δ(x) The flowing layer has shape δ(x) = δ 0 1 x 2 /L 2

10 Constant rotation rate At constant rotation rate particle streamlines form closed loops passing through flowing layer steady, divergence-free, integrable can transform to action angle coordinates ρ, φ trajectories in action angle coordinates given by: ρ = 0 φ = 2π/T (ρ) taking a time τ-map gives a twist map P(ρ, φ) = (ρ, φ + 2πτ/T (ρ))

11 Variable rotation rate Break the integrability by varying the rate of angular rotation Sinusoidal forcing has been well-studied. [Fiedor and Ottino, JFM ]

12 Variable rotation rate [Fiedor and Ottino, JFM ]

13 Variable rotation rate Key idea is that streamlines changes and cross [Fiedor and Ottino, JFM ]

14 Piecewise constant rotation rate Simplify the forcing by using a blinking flow ω b = ω + ˆω for iτ < t < (i + 1/4)τ ω = ω a = ω ˆω for (i + 1/4)τ < t < (i + 3/4)τ ω b = ω + ˆω for (i + 3/4)τ < t < (i + 1)τ Alternate the angular velocity between ω a and ω b.

15 Poincaré sections N = 2 N = 4 N = 6 N = 8

16 Blinking experiments

17 Linked Twist Maps on the plane Domain is two intersecting annuli with two distinct regions of intersection

18 Linked Twist Maps on the plane The action of a twist map is to take a line...

19 Linked Twist Maps on the plane... and twist it around the annulus. Then do the same with points in the other annulus. Proof of ergodic mixing due to Burton & Easton (1980), Devaney (1980), Wojtkowski (1980), Przytycki (1983)

20 Linked Twist Maps on the plane A twist map takes points in an annulus...

21 Linked Twist Maps on the plane... and performs a shear, wrapping this initial set around the annulus F(r, θ) = (r, θ + f (r)) (centred at the centre of left annulus)

22 Linked Twist Maps on the plane A linked twist map is the composition G F of such maps on a pair of annuli. F(r, θ) = (r, θ + f (r)) (centre left annulus) G(ρ, φ) = (ρ, φ + g(ρ)) (centre right annulus) Proof of ergodic mixing due to Burton & Easton (1980), Devaney (1980), Wojtkowski (1980), Przytycki (1983)

23 Linked Twist Maps on the plane

24 The Blinking Vortex

25 The Blinking Vortex

26 The Blinking Vortex

27 The Blinking Vortex

28 Streamline crossing structure

29 3-dimensional spherical tumbler An obvious way to introduce some transversality...

30 Rotation about the z-axis Solid body rotation in the bulk: ẋ = ωy ẏ = ωx ż = 0 Shear in the flowing layer: ẋ = γ 1 (δ 1 (x, z) + y) ẏ = ω 1 xy/δ 1 (x, z) ż = 0 Boundary of flowing layer and bulk: δ 1 (x, z) = δ 0 1 x 2 /L 2 = ω 1 /γ 1 R 2 x 2 z 2

31 Rotation about the x-axis Solid body rotation in the bulk: ẋ = 0 ẏ = ωz ż = ωy Shear in the flowing layer: ẋ = 0 ẏ = ω 2 zy/δ 2 (x, z) ż = γ 2 (δ 2 (x, z) + y) Boundary of flowing layer and bulk: δ 2 (x, z) = δ 0 1 z 2 /L 2 = ω 2 /γ 2 R 2 x 2 z 2

32 Dynamical properties I Theorem If ω 1 /γ 1 = ω 2 /γ 2 then motion of a particle is constrained to a single hemispherical surface.

33 Dynamical properties II Theorem Period 1 points for rotation about the x-axis form a bowl in the shape of a prolate spheroid.

34 Intersection of bowl and spheroid

35 Experiments Experiments... are hard how do you see what s happening inside a sphere filled with a granular material?

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39 Piecewise isometries [Goetz, 2003]

40 Properties of piecewise isometries Is piecewise isometric dynamics chaotic? They have sensitive dependence on initial conditions On the other hand, topological entropy is zero [Buzzi, 2001] Orbits do not diverge exponentially, but in polynomial order Sometimes referred to as weakly chaotic or pseudo-chaotic.

41 Dynamics of piecewise isometries

42 Dynamics of piecewise isometries Piecewise isometries can possess efficient mixing behaviour in the absence of any stretching and folding.

43 Comparison with non-zero flowing layer Piecewise isometry Fast flowing layer Realistic flowing layer

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46 Conclusions and questions Segregation frequently dominates granular media, and rotations about different axes offers an opportunity to produce mixing in the absence of stretching. What significant (robust) features of PWIs can we expect to observe in granular experiments? Is there a systematic understanding of PWIs as a limiting behaviour of a continuous system? How do the PWI dynamics compete with shearing from the flowing layer, and with segregation effects?